Saturday, March 27, 2010

A Splash or Two of Precognitive Dreaming

The Persistence of Memory - Salvador Dali

Today, when I got up, I commented to my partner that I had had a bizarre dream that I was back in in a Maths Dept meeting, although I have now retired.

At lunch time I sent my coffee cup flying by accidentally poking my fingers into the rim as I was turning at the table, nearly drenching the keyboard and splashing myself down the trousers, with an uncanny feeling this was becoming a bad habit.

The next cup of green tea, which I have with orange juice to enhance the anti-oxidant epicatechin gallate, I caught the orange juice container, which had a loose lid, against the fridge door and sent the lid flying, scattering juice over the floor, but caught the container in mid-air. Then suddenly it all came back to me!

In the middle of the dream, in which I was making comments which everyone was reluctantly tolerating, because I was retired and shouldn't have an opinion anyway, (actually it was weirder than that - I was talking in a chirpy voice and they couldn't hear me as if I was a ghost - which is a signal of the deeper dreaming stratum), anyway in one of those dream disconnections, this was suddenly interrupted by me getting up scattering my coffee cup on the floor, when I stabbed my finger accidentally into the rim! Not only that, but when I went to bend over to try to pick it up, the entire coffee urn, which was a large chromed boiler on three legs, fell all over the floor as well and couldn't be uprighted because the legs had become rubbery wires, dysfunctional, like the cardboard orange juice container with the unglued screw cap.

So the dream has two splashed containers just as the day has, I reported the dream occurrence hours before the events happened, and it was only when the second splashing became real that the whole weight of the four mistakes reminded me of the full content of the dream and the uncanny implications ...

This harks back to the first precognitive dream I had when I dreamed two nightmares that I was being stung, told my young wife as she arose to feed our newborn daughter, and an hour later was stung wide awake in bed by a wasp that had flown in the window my wife opened after she got up.

My son's response was "of course it might be that the dream primed the events", but this is as much a violation of causality as precognition. Both the real events were accidents, not intentional acts, so priming suggests that dreaming something, particularly in these deep intense hallucinogenic uncontrolled phases of dreaming, might affect future outcomes, not just cause an imprint leading to the person acting out the dream later.

This is of course what quantum theories of dreaming and consciousness might be about, namely anticipating future events which could have significance for the survival of the experiencer. If one dreams a plane is about to crash and one is about to fly, perhaps the dream is priming towards a set of probability futures, or parallel universes, in which these experiences have a non-zero probability of occurring. The dream then leads to a capacity on the part of the experiencer to anticipate disaster and hopefully to evade annihilation.

If so, this anticipatory capacity is not necessarily confined to dreams, but may be the founding property which has caused brains to evolve to become subjectively conscious, namely to anticipate chance happenings beyond the organisms control - perhaps in waking life just in the immediate future of the 'quantum of the present' enabling one, through a degree of foreboding to jump out of the way of the tiger just in time, or on a slightly longer-term basis to intuitively help us decide which shady path ahead the tiger might actually be on.

This would appear to link the unstable and unpredictable properties of a brain state critically poised in self-organized criticality of edge-of-chaos dynamics to respond to quantum sensitivity of its own butterfly effect to generate a form of quantum entanglement that becomes sensitive to the potential future states of the whole system. This would then become the crack in the cosmic egg, in which consciousness has room to utilize Carlos Castaneda's cubic centimeter of chance to good effect.

Finally here is one that happened a couple of months later that shows the same reporting veracity as the stung dream. Last night I dreamed that a group of women had a newborn baby together and they turned to me and threw it out of their van, claiming they had had trying to look after it, claiming was actually mine. I caught it in mid-air and went to look after it but it had strange glutinous transparent white bowel motions. When I woke up I told Christine about it. Then tonight we were watching "House" and the first episode was "Maternity". It began with a couple with a newborn baby which then began to burp up its milk and had an intestinal blockage. Then it became a viral epidemic and second lesbian couple lost their baby. So there were the two elements together in the dream, a women-only group with a baby losing it and a baby with intestinal troubles emitting white glutinous liquid.

Wednesday, March 24, 2010

Reproductive Passion, Tantric Completion and Polyamory

My second menage-a-trois was a time of both sexual adventure and domestic bliss.

Sexual Paradox - our website

Polyamory as the Spice of Fidelity

I was a shy, only child become a willowy male who met a mysterious beatnik American expatriate girl who was way ahead of me in the love stakes, having already had a string of heady relationships and sexual experiences, when I was still a gangling somewhat jealous virgin.

We married on the somewhat paradoxical basis that she could have other lovers should she wish later - an option encouraged by her father. Things were relatively idyllic for three years and we had our first two offspring, a son and a daughter.

Then she fell in love with one of our best friends. she was the experienced, confident, independent party and I was the anxious, dependent, paranoid party and it dragged our relationship down to the point where we split up for a short period, just as I began to have my first exploratory polyamorous relationship with a younger music student, who was a brilliant pianist.

My first double partners

Then the three of us came together again partly as a way of keeping the children together and looking after them as a family and we lived for a year and a half as a menage-a-trios. This led to a crazy period in the seventies where we held open free love house to a number of our hippie friends and pursued an ever increasing number of casual and intimate relationships, in a kind of race towards the full sexual charisma of loving or being loved by just about everyone we knew. Those we the innocent days when the worst disease we suffered was genital herpes.

It was at this point that I met Christine and she became an integral partner in my second double marriage, and we proceeded to have one, and later two, children together. My wife as the 'leading lady' was highly ethical about her relationships with both the other women and relations were friendly and conducive to a resourceful and positive family environment for our children. Four and a half happy adventurous, but chaotic, years ensued in relationship terms, in which we brought up the children together while pursuing a variety of strange affairs that eventually led to the breakup of all three of us.

My wife became involved with another of our close associates and became pregnant to him with whom she later had a second child all of whom have become an integral part of our extended family of adult offspring. Christine also became involved with another associate, just as I was heading off alone on my first sabbatical to wander India as a sadhu, and do research in Europe and the US - the ultimate full circle back to solo existence.

The simultaneous split ups with my two familial partners were very painful, but we agreed above all to look after the children together, with them able to come and go freely between us on neighbouring blocks of land, on an equal time-shared basis, after I returned from my year overseas. This kept the overall family intact and enabled it to become an even larger extended family, all of whom share a deep bond of love and commitment to one another, even to the third generation.

I continued to be polygynous by choice and entered into a third double relationship for 15 years or so with Christine and another partner. This was also fraught with tension caused by the two partners' reluctant and at best partial acceptance of one another, even when we sometimes had other male partners in common, leading after a couple of years to each having separate domains with me going to and fro between. This is an endless problem of polygynous marriages.

Eventually, after Christine and I had my fourth child together, my relationship with my other partner ended in overt terms, only to become a long drawn out smouldering clandestine affair - a decision I made after continuing tension, made to confirm my commitment to the partner with whom I was rearing a family, while still being faithful to the principle of being lovers with my other partner, as opportunity provided.

Eventually this situation led to separation, but was followed by a consensual open partnership with Christine, where it was possible to negotiate to have an outside sexual relationship, but only if ring fenced with total care for our health, by having 'pre-nuptial' STD tests of any potential outside partners to protect existing partners who could be harmed. This was an essential reaction to the HIV era and something we have never been prepared to compromise on since for all the casual habits of swingers, the tortured lifestyle choices of devoted polyamorists, let alone the unprotected clandestine affairs of more conventional people.

I have since had a small number of outside relationships to further a creative psychic and physical relationship with another person, not for the fulfillment of polyamorous sexual desires, but to live out fully an intimate creative relationship with a member of the opposite sex with whom I have had an emerging engagement.

The critical point in favour of open polyamory is that it makes it possible to be honest that you are entertaining an affair and to gain some acceptance of it from ones existing partner, establishing some trust around what is often a painful situation if only one partner is actively seeking a new relationship. Conventional monogamy has the almost inescapable pitfall that it leads to deceit and betrayal in the affairs of love and often the health of the affected parties, leading to divorce settlements, and acrimonious contests over custody.

Finally, for the last seven years, Christine and I have entered into a period of creative monogamy, in which we are each giving our full energy to our sexual relationship with one another and making it into a passionate saga of unforgettable experiences, whose ultimate peaks we are continually in the process of scaling to mutual oblivion. This has proved for both of us in our sixties to be the unleashed sexual high point of our whole lives. I am still inherently polygynous, as most men are liable to sew wild oats by evolutionary trait, but I sublimate this, through using mild pornography as Tantric exercises towards another bout of creative sexual reunion with the beloved.

Reproductive Passionate Love as the Holy Grail of Eros

We have a highly heterosexual view of sexuality that prizes Tantric complementation of female and male energies as manifesting the full completion with the alien exotic other. This is not just sex as erogenous gratification, but more of a cosmic fusion and annihilation, in which both the woman and the man play an active creative part in the inner mounting flame of love, as forthright as Krishna is to Radha and as topsy turvy as Kali is to Shiva.

Willendorff Venus meets Green Man

It also holds, as its full quintessence, reproductive fertilization, and the implicit capacity of sexuality to emanate the life force both psychically, energetically and in the capacity to get physically pregnant and to give birth to new life.

It worships the sexual differences between the two sexes, as fully manifesting the complementary nature of the Tao in its totality between wave and particle, mind and body, female and male, chaos and order and so on.

While it is true that many of the Tantric temples of India and Nepal, such as a the Lakshman temple at Khajuraho, have lower freezes depicting all the sexual postures and erotic perversions of which humanity is capable, from bestiality to anal and oral sex, the higher pinnacles depict the union of Shakti and Shiva, not just of two unisex people, gay, lesbian or coincidentally heterosexual, performing any act from sodomy to cunnilingus without discrimination, but the intimate complementarity of female and male cosmic principles, in reproductive lingam-yoni coitus, often surrounded by a supporting group of attendants.

Khajuraho upper and lower friezes

In today's epoch of consenting adult eros and the pursuit of simple gratification by any means, there is only a limited difference between heterosexual and homosexual eroticism because both engage the same spectrum of perverse erotic acts unrelated to the reproductive quest. There is a poetic irony in that heterosexual pornography is as engaged as gay male voyeurism in non-reproductive sexuality, from exploitative deep-throat fellatio to sometimes violent sodomy, as an expression of male sexual power.

It may well be that this is ancient - that some women throughout history have been prepared to subject themselves to any form of sexual concourse to give their man everything he desires and thus to hold on to him by letting him possess them utterly and completely, but it is rarely for their own sexual fulfillment or sexual autonomy that they do this. Female eros and particularly astute female reproductive choice takes a very different direction of its own, more likely to very discretely seduce and attract the most genetically endowed and capable man in sight on her own terms of ecstatic enticement.

Eros only makes sense if integrated into creative relationship with the deeper unifying truth of the fusion of the complementary principles of which the universe is composed in the burgeoning of the diversity of new life. This doesn't mean neglecting contraception, or denying women the right to reproductive choice. In fact it is founded on the principle that women, as the principal bearers of new life, must have reproductive sovereignty over their own autonomy and a degree of sexual choice that can't be made subject to male control.

The strategic preservation of female reproductive choice is why females have evolved to possess an ecstatic orgasm even though it is unnecessary for fertilization, why ovulation is concealed, and why women have become sexually attractive at all phases of their reproductive cycle.

It is because my beloved partner has pendulant flowing breasts which I have seen and tasted running with milk and an aromatically delightful copulin-filled vagina and I am fertilizing her fullness with my manhood to my oblivion with all the poly-orgasmic manifestations of the abundance of life flowing forth that affirm the sexual complementation is complete. It is also because the sexuality is implicitly reproductive that it fosters long-lasting meaningful relationships, related to family, the passage of the generations and thus the immortality of ongoing life.

Thy teeth are like a flock of sheep that are even shorn, which came up from the washing;
whereof every one bear twins, and none is barren among them.

It is not self-love, or the love of self-natured sexuality, which is forever limited, because it ultimately seeks sameness and is thus narcissistic, even when one or other in a uni-sex relationship assumes the stylized roles of butch or doe, but the worship of the complete other in the overflowing fullness of their natural abundance.

The seeking of the complementary 'other' is at the evolutionary source of all sexuality, because in sexual reproduction we communicate only half our genes in an intermingling with the other's, and it is through the genetic differences between the two parents that each generation gains its adaptive strength and resistance to disease. This is why women are pheromonally attracted to men with complementary histocompatibility genes and resist the incestuous advances of close male relatives.

It is also a completion, by contrast with the sexually stereotypic extremes of male-exclusive or female exclusive sexuality, which despite its pretensions to a more overarching gender encompassing identity, tends to extremes of runaway male promiscuity or female nesting.

"There's a joke in the gay community that goes like this -
'What does a lesbian bring on her second date?' - 'A U-Haul'.
'And what does a gay man bring on his second date?' - "What second date?'

Reproductive passionate love is not in any way homophobic - it has no need to need to be for the very reason that it is complete, but it IS all-consumingly and overflowingly heterophilic, as Tao and Tantra, and as the Holy of Holies - the Song of Songs declare:

Open to me, my sister, my love, my dove, my undefiled:
for my head is filled with dew, and my locks with the drops of the night.
My beloved put in his hand by the hole of the door, and my bowels were moved for him.
I rose up to open to my beloved; and my hands dropped with myrrh,
and my fingers with sweet smelling myrrh, upon the handles of the lock.

Sunday, March 21, 2010

Primes out of Thin Air

Left: von Mangoldt additive prime counting function generated using the explicit formula over the Riemann zeta zeros.
Right: Fourier sine transform of the additive von Mangoldt function showing coincidence with the zeta zeros.

The Riemann zeta zeros behave very similarly to a kind of Fourier transform of the primes. This led to the idea of performing a Fourier sine transform of the additive Mangoldt function which is the subject of the explicit prime formula. In his AMS paper Brian Conrey illustrates this process, which in a personal e-mail he attributes to Mike Rubinstein with Mike's quotes "I made the picture to illustrate that one could easily detect the zeros just from numerical data, looking at it on a logarithmic scale."

While writing my research review of the Riemann hypothesis and zeta fractal dynamics I began playing around with alternatives to the von Mangoldt function in the Fourier transform which might give rise to some other zeta-like zero distributions, I accidentally programmed in a simple step function and found it gave a highly smoothed transform, which still had irregular characteristics and a close correspondence to the zeta zeros, from which it was much easier to extract meaningful local minima than the original von mangoldt transform above.

Having got a very close correspondence, with only minor variations, to the zeta zeros, running in a consistent sequence for hundreds of values, I decided to explore what would happen if these values were chosen on the critical line and applied to the explicit formula generating prime counting from zeta zeros (complex value = rho):

To my fascination, the minima produced a meaningful integer-step pseudo-prime distribution, which showed both relative maxima at integer values, as would have perhaps been expected if the explicit formula were to produce any sort of meaningful result, which is itself surprising, but also, reflecting the zeta-approximate irregularities in the transform, it appears to have consistent semi-local peaks at primes and prime powers. So we have generated a distribution back out of the Fourier transform that intimates the primes out of thin air, in the sense that the prime counting function has been removed entirely from the Fourier transform.

Click to enlarge: (Top left) Fourier sin transform of integer step function (black) overlaid on that of the prime counting function (green) and actual zeta zeros (blue). (Top right) Step function minima distribution blue closely coincides with zeta zero distribution (green). (Lower right) Difference between the two. (Lower left) Applying the explicit formula to the step minima at real part 1/2 results in a function with integer local maxima with neighbourhood peaks at primes. (Inset) The step function and summed Mangoldt prime distribution compared.

Anyone familiar with the relationship between the Euler product over primes and the zeta function as a sum
will realize immediately that a similar Euler product over integers would correspond to a sum whose coefficients, instead of being 1, would be the rapidly increasing number of possible factorizations of n, unlikely to give rise to a convergent function with an analytic extension. This makes the fact that the explicit formula does give a meaningful distribution from the integer step function all the more surprising.

Of course the challenge is to you to find if there is a glitch in the calculation causing this phenomenon. I presume it is because the Fourier transform is performing a subtle version of prime sieving, just as the zeta function does with the Euler product. So you can verify it on the spot, I have included the Matlab m-file generating the process below.

function fintstep(scale,start,fin,its,xres)
% fintstep(10^6,1,50,490,600);
% plots Fourier sin transform of integer step distribution giving minima

% coinciding closely with zeta zeros

% 1/log(X) int_0^log(X) sin(ty) (stp(e^y)-e^y)/e^(y/2) dy
% parameters start = first prime, fin = last prime,
its = how many zeta zeros to use xres = number of x points in graph
% sav will save the arrays z, x if needed prtits=1 print iterations

for j=2:scale;
%make the additive Mangoldt prime counting function
for t=1:xres
for yy=0:1/stpval:log(X)

'g'); %first figure plots Fourier sin transform
%additive Mangoldt transform in green
for j=2:scale;
y(j)=y(j-1)+1; %make the integer step function
for t=1:xres
for yy=0:1/stpval:log(X)

'k'); %integer step function transform in black
for j=1:100 %also plot in blue vertical lines at the actual zeta zeros
if zros(j)>myspan/10 break;
else line([zros(j) zros(j)], [-500 500]);

%making a vector out of the local minima of the Fourier transform
for k=2:length(F)-1;
if F(k)<=F(k-1)&&F(k)<=F(k+1) x=[x xx(k)]; end
%xzeros becomes the vector of minima
%plot step zeros in blue compared with zeta zeros in green
if its>length(zros)
for l=1:its
for j=1:xres
%pair off zeros above and below (conjugates)
if l>1
y(l,j)=y(l,j)+y(l-1,j); %sum the contributions from all the zeros
%the explicit prime formula
for j=1:length(p)
if (p(j)<=fin) line([p(j) p(j)],[-10 80]); end

%von Mangoldt function %o=log(p) if n=p^k or 0 otherwise
if f(1)==f(length(f))

Newton Gliding into Riemann's Zeta

This blog is an illustration of how it becomes possible to model the most dizzyingly complicated complex functions known in rich detail highlighting the zeros of Riemann's zeta and Xi functions.

Newton's method finds the zeros of a function by iteratively running up to the graph of the function (green) and then down the slope of the function (red) and repeating the process, converging towards a zero of the function.

This follows the formula below defining the Newton function Nf of a function f. For example, if we take the complex function f(z)=z3-1, whose zeros are the complex cube roots of unity, we have the following iterations.

When we carry out this iteration in complex numbers, we find that some points don't tend to the zeros and instead enter a chaotic state, becoming the fractal Julia set of the iteration illustrated below.

The three red, brown and green fractal basins are the points iterating to each of the three zeros, but the points on the black skeleton in the inset are on the chaotic Julia set of the iteration.

Notice that every point on the Julia set bounds all three basins, so this is a fractal landscape involving three regions in which every point on any of their boundaries bounds all three regions, a seeming topological impossibility unless the set is fractal.

What then happens if we try to investigate the most complex and enigmatic function of all, the Riemann zeta function, whose zeros seem to form a Fourier transform, or hologram, of the prime numbers?

The following set of equations show how we have to work a 'Houdini act' to define zeta, and extend it firstly from Re(z)>1 down to Re(z)>0 and then over the entire complex plane except the singularity at z=1, using the gamma function and how we can in turn approximate the gamma function to calculate zeta.

But to get the Newton function of Riemann's zeta, we have to first differentiate it, which requires even more equation subtlety and a lot more processing power.

Thus we finally arrive above at a workable formula for Newton zeta and the related Newton Xi which also has the same enigmatic set of zeros.

(Click to enlarge)

Above we see at left the zeta function with its zeros, next the Newton function of zeta, attracting the non-trivial zeros on the line x=1/2 in green and the trivial zeros on the negative real axis at x=-2n in red. Notice the fractal way in which satellites of the basins are entangled.

Finally above center we have the Newton function of Xi with the basin of the third non-trivial zeta zero in red and the basins of the others in green, showing how the basins of the zeros are fractally reflected in one another, as shown in the two exploded images at right of fractal neighbourhoods of ensuing boundaries between the zeros immediately above and below the third one.

This has all been made possible thanks to the GCC complex.h math library, and the XCode integrated development environment for Mac in generating the applications downloadable in the previous blog.